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Efficient and Scalable Graph Generation through Iterative Local Expansion

Andreas Bergmeister, Karolis Martinkus, Nathanaël Perraudin, Roger Wattenhofer

TL;DR

This work tackles the scalability gap in graph generation by introducing iterative local expansion, which grows graphs from a single node and refines local structure via diffusion. It combines a coarsening-inversion framework with a diffusion-based denoiser, a locally expressive Local PPGN, and spectral conditioning to preserve global properties while maintaining subquadratic runtime for sparse graphs. The method achieves state-of-the-art or competitive results on standard benchmarks, scales to graphs with thousands of nodes, and uniquely extrapolates to out-of-distribution sizes while retaining key structural characteristics. These contributions enable robust, scalable graph generation with practical applicability to large, real-world graphs.

Abstract

In the realm of generative models for graphs, extensive research has been conducted. However, most existing methods struggle with large graphs due to the complexity of representing the entire joint distribution across all node pairs and capturing both global and local graph structures simultaneously. To overcome these issues, we introduce a method that generates a graph by progressively expanding a single node to a target graph. In each step, nodes and edges are added in a localized manner through denoising diffusion, building first the global structure, and then refining the local details. The local generation avoids modeling the entire joint distribution over all node pairs, achieving substantial computational savings with subquadratic runtime relative to node count while maintaining high expressivity through multiscale generation. Our experiments show that our model achieves state-of-the-art performance on well-established benchmark datasets while successfully scaling to graphs with at least 5000 nodes. Our method is also the first to successfully extrapolate to graphs outside of the training distribution, showcasing a much better generalization capability over existing methods.

Efficient and Scalable Graph Generation through Iterative Local Expansion

TL;DR

This work tackles the scalability gap in graph generation by introducing iterative local expansion, which grows graphs from a single node and refines local structure via diffusion. It combines a coarsening-inversion framework with a diffusion-based denoiser, a locally expressive Local PPGN, and spectral conditioning to preserve global properties while maintaining subquadratic runtime for sparse graphs. The method achieves state-of-the-art or competitive results on standard benchmarks, scales to graphs with thousands of nodes, and uniquely extrapolates to out-of-distribution sizes while retaining key structural characteristics. These contributions enable robust, scalable graph generation with practical applicability to large, real-world graphs.

Abstract

In the realm of generative models for graphs, extensive research has been conducted. However, most existing methods struggle with large graphs due to the complexity of representing the entire joint distribution across all node pairs and capturing both global and local graph structures simultaneously. To overcome these issues, we introduce a method that generates a graph by progressively expanding a single node to a target graph. In each step, nodes and edges are added in a localized manner through denoising diffusion, building first the global structure, and then refining the local details. The local generation avoids modeling the entire joint distribution over all node pairs, achieving substantial computational savings with subquadratic runtime relative to node count while maintaining high expressivity through multiscale generation. Our experiments show that our model achieves state-of-the-art performance on well-established benchmark datasets while successfully scaling to graphs with at least 5000 nodes. Our method is also the first to successfully extrapolate to graphs outside of the training distribution, showcasing a much better generalization capability over existing methods.
Paper Structure (57 sections, 33 equations, 13 figures, 7 tables, 8 algorithms)

This paper contains 57 sections, 33 equations, 13 figures, 7 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Method
  4. Graph Expansion
  5. Probabilistic Model
  6. Factorization
  7. Learning to Invert Graph Coarsening
  8. Contraction Families
  9. Variational Interpretation
  10. Spectrally Guided Generation
  11. Modeling and Training
  12. Modeling with Denoising Diffusion Models
  13. Local PPGN
  14. Spectral Conditioning
  15. Perturbed Expansion
  16. ...and 42 more sections

Figures (13)

  • Figure 1: Example of a 4-level coarsening sequence. Colors indicate the node contraction sets ${\mathcal{V}}^{(p)}$. Our generation process aims at reversing with expansions and refinements the $T$ steps of this sequence from $G_T$ to $G_0$. The details of a single step are provided in Figure \ref{['fig:single-step']}.
  • Figure 2: Single step schematic representation of the proposed methodology. The upper row delineates two sequential coarsening steps, using color differentiation to denote the contraction sets ${\mathcal{V}}^{(p)}$. Commencing from the right in the lower row, the expansion of $G_{l+1}$ into $\tilde{G}_{l} = \tilde{G}(G_{l+1}, {\bm{v}}_{l+1})$ is shown, assuming a known cluster size vector ${\bm{v}}_{l+1}$. Colors distinguish membership within expansion sets while dashed lines indicate edges to be removed as per the edge selection vector ${\bm{e}}_{l}$. The resultant refined graph $G_{l} = G(\tilde{G}_{l}, {\bm{e}}_{l})$ is shown in the central box, where node features correspond to the cluster size vector ${\bm{v}}_{l}$, used in expanding $G_{l}$ into $\tilde{G}_{l-1}$ (illustrated in the leftmost box).
  • Figure 3: Extrapolation and interpolation to out-of-distribution graph sizes. The shaded area represents the training size range.
  • Figure 4: Depiction of a perturbed expansion. The graph $G_l$ is expanded into $\tilde{G}_{l-1}$ utilizing the cluster size vector aligned to the node features. Deterministic expansion components are represented by linear black edges, whereas curved red edges showcase supplemental edges implemented for a radius of $r=2$ and a probability of $p=1$. With $p<1$, a subset of these edges would be randomly excluded.
  • Figure 5: Illustration of a directed graph with self-loops. In the following, we list the update formula for three representative edge embeddings: $\textcolor{#26FF52}{({\bm{h}}')^{(1,1)}} = \gamma \left( \textcolor{#26FF52}{\bm{h}}^{(1,1)}, \phi \left(\textcolor{#26FF52}{\bm{h}}^{(1,1)}, \textcolor{#26FF52}{\bm{h}}^{(1,1)} \right) \oplus \left(\textcolor{#FFD966}{\bm{h}}^{(1,2)}, \textcolor{#FF0000}{\bm{h}}^{(2,1)} \right) \oplus \left(\textcolor{#67AB9F}{\bm{h}}^{(1,3)}, \textcolor{#CC99FF}{\bm{h}}^{(3,1)} \right) \right)$$\textcolor{#67AB9F}{({\bm{h}}')^{(1,3)}} = \gamma \left( \textcolor{#67AB9F}{\bm{h}}^{(1,3)}, \phi \left(\textcolor{#26FF52}{\bm{h}}^{(1,1)}, \textcolor{#67AB9F}{\bm{h}}^{(1,3)} \right) \oplus \left(\textcolor{#FFD966}{\bm{h}}^{(1,2)}, \textcolor{#6600CC}{\bm{h}}^{(2,3)} \right) \oplus \left(\textcolor{#67AB9F}{\bm{h}}^{(1,3)}, \textcolor{#CC99FF}{\bm{h}}^{(3,1)} \right) \right)$$\textcolor{#0000FF}{({\bm{h}}')^{(3,4)}} = \gamma \left( \textcolor{#0000FF}{\bm{h}}^{(3,4)}, \phi \left(\textcolor{#0000FF}{\bm{h}}^{(3,4)}, \textcolor{#66B2FF}{\bm{h}}^{(4,4)}\right) \right)$
  • ...and 8 more figures

Theorems & Definitions (5)

  • Definition 1: Graph Expansion
  • Definition 2: Graph Refinement
  • Definition 3: Graph Coarsening
  • Definition 4: Perturbed Graph Expansion
  • Definition 5: Restricted Spectral Approximation